#470969

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,19,15,21,30,20)(11,12,24,16,17,27,23,26)(13,28,31,38,32,18,35,37)(22,39,36,29,34,25,40,33);(1,3,4,7)(2,5,6,8)(9,11)(10,16,21,26)(12,20,27,19)(13,22)(14,23)(15,17)(18,33,28,29)(24,30)(25,38,39,37)(31,40)(32,34)(35,36);(9,12,14,16,15,27,30,26)(10,17,20,24,21,11,19,23)(13,29,31,25,32,33,35,39)(18,36,38,22,28,40,37,34);(2,6)(5,8)(9,13,30,35,15,32,14,31)(10,18,20,38,21,28,19,37)(11,22,24,36,17,34,23,40)(12,25,16,33,27,39,26,29)

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378165);